Integrating Other Functions (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

Integrating other functions

Is integration the reverse of differentiation?

• Yes, but remember “+c”, the constant of integration …

  • … unless finding a definite integral

• Recognising common results helps to make integration easier

  (See Differentiating Other Functions)

• You are given all the standard integrals in the formula booklet

How do I integrate e^x and e^kx?

• The gradient of e^kx is ke^kx

  • ie y = e^kx, dy/dx = ke^kx

• The reverse applies when integrating

• This is an example of reverse chain rule

How do I integrating 1/x?

• Remember 1/x = x^-1

  • The method for integrating powers does not apply if the power is -1

How do I integrating sin x and cos x?

• Note the minus in the integral of sin x

How do I integrate tan x?

• The integral of tan x is ln|sec x| + c

How do I integrate $\frac{1}{1+x^2}$ and $\frac{1}{a^2+x^2}$?

• Recall that $\frac{\mathrm{d}}{\mathrm{d}x}\left({\tan }^{-1}x\right)=\frac{1}{1+x^2}$

• Then $\int \frac{1}{1+x^2}\mathrm{d}x={\tan }^{-1}x+c$ and $\int \frac{1}{a^2+x^2}\mathrm{d}x=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+c$